Newcomb's Paradox

Newcomb’s paradox was conceived by the physicist William Newcomb in 1960, and it explores the perennial philosophical problem of free will versus determinism. Here is the basic setup:

There are 2 boxes labeled A and B sitting on a table in a room. Box A is transparent and you can see that it contains $1,000 dollars. Box B is opaque and you can’t see inside. Now you are given a choice. You can take both boxes A and B, or only box B. However, you are told that before you entered the room, a supreme being who can make highly accurate predictions has done one of two things. If the being predicted that you would choose only box B, then they placed one million dollars in it. If they predicted that you would take both boxes A and B, they left box B empty.

You have to make the decision on your own, and you cannot use an outside device to randomize your decision (e.g. flipping a coin). You also trust that the supreme being can indeed make highly accurate predictions about the future. What choice would you make?

When I first encountered this problem my inclination was to take only box B. After all, it's not every day that you get an opportunity to win one million dollars. However, the supreme being has already made their choice, and box B either contains one million dollars or it doesn't. Taking both boxes wouldn't suddenly cause the money to disappear if it is already in there. Therefore, why not take both?

The “paradox” is that a convincing argument can be made for either choice.

If you are skeptical of the supreme being's predictions, consider the following slightly altered version.

The scenario remains the same, but let's assume that it has been repeated multiple times with different people before you. In every case, when the person picked both boxes, box B was empty. When they chose only box B, it contained one million dollars. So the supreme being's forecast was consistently accurate. You have no reason to believe that your situation will be any different. In light of this evidence, it would be foolish not to choose box B, wouldn't it?

However, let's assume that you have a friend seated behind box B, and they can see what's inside it. Although your friend cannot convey any message to you, they can see whether box B is empty, or contains one million dollars. Clearly, your friend wants you to take both boxes because at worst you'll end up with $1,000, and at best with $1,001,000. Why not leverage the fact that the supreme being has already made their move?

As far as I know, there is no widely accepted “solution” to Newcomb’s paradox. Even when modeled from a rigorous game theoretic perspective, there are reasonable assumptions that can lead to either conclusion.

The venerable Martin Gardner, who gives an excellent account of Newcomb’s paradox in his Colossal Book of Mathematics, points out that whenever a prediction interacts with the predicted event, logical contradictions are bound to arise. In this case, the interaction happens when you are told that your choice has already been predicted. Gardner goes on to say that these types of situations strike at the heart of free will.

Consider this simple scenario. The supreme being predicts that when you go to bed tonight, you will take your shoes off. If you are never made aware of this prediction, there is no problem; but if the supreme being informs you of the prediction, then you can easily falsify it by sleeping with your shoes on.

In any case, Newcomb's paradox is a fun thought experiment to pose to a class of students, or to your extended family if you'd like to argue about something other than politics at your next gathering.